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1.
A theorem of Hardy characterizes the Gauss kernel (heat kernel of the Laplacian) on ℝ from estimates on the function and its Fourier transform. In this article we establisha full group version of the theorem for SL2(ℝ) which can accommodate functions with arbitraryK-types. We also consider the ‘heat equation’ of the Casimir operator, which plays the role of the Laplacian for the group. We show that despite the structural difference of the Casimir with the Laplacian on ℝn or the Laplace—Beltrami operator on the Riemannian symmetric spaces, it is possible to have a heat kernel. This heat kernel for the full group can also be characterized by Hardy-like estimates.  相似文献   

2.
We prove two versions of Beurling's theorem for Riemannian symmetric spaces of arbitrary rank. One of them uses the group Fourier transform and the other uses the Helgason Fourier transform. This is the master theorem in the quantitative uncertainty principle.

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3.
We consider tight Gabor frames (h,a=1,b=1) at critical density with h of the form Z −1(Zg/|Zg|). Here Z is the standard Zak transform and g is an even, real, well-behaved window such that Zg has exactly one zero, at , in [0,1)2. We show that h and its Fourier transform have maximal decay as allowed by the Balian-Low theorem. Our result illustrates a theorem of Benedetto, Czaja, Gadziński, and Powell, case p=q=2, on sharpness of the Balian-Low theorem.   相似文献   

4.
Out problem is about propagation of waves in stratified strips. The operators are quite general, a typical example being a coupled elasto-acoustic operator H defined in ?2 × I where I is a bounded interval of ? with coefficients depending only on zI. The “conjugate operator method” will be applied to an operator obtained by a spectral decomposition of the partial Fourier transform ? of H. Around each value of the spectrum (except the eigenvalues) including the thresholds, a conjugate operator is constructed which permits to get the ”good properties“ of regularity for H. A limiting absorption principle is then obtained for a large class of operators at every point of the spectrum (except eigenvalues).  相似文献   

5.
First, the basic concept of the vector derivative in geometric algebra is introduced. Second, beginning with the Fourier transform on a scalar function we generalize to a real Fourier transform on Clifford multivector-valued functions Third, we show a set of important properties of the Clifford Fourier transform on Cl3,0 such as differentiation properties, and the Plancherel theorem. Finally, we apply the Clifford Fourier transform properties for proving an uncertainty principle for Cl3,0 multivector functions.  相似文献   

6.

We study fundamental properties of product (α1, α2)-modulation spaces built by (α1, α2)-coverings of ℝn1 × ℝn2. Precisely we prove embedding theorems between these spaces with different parameters and other classical spaces. Furthermore, we specify their duals. The characterization of product modulation spaces via the short time Fourier transform is also obtained. Families of tight frames are constructed and discrete representations in terms of corresponding sequence spaces are derived. Fourier multipliers are studied and as applications we extract lifting properties and the identification of our spaces with (fractional) Sobolev spaces with mixed smoothness.

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7.
This paper studies the structure of shift-invariant spaces. A characterization for the univariate shift-invariant spaces of tempered distributions is given. In Lp case, an inclusive relation in terms of Fourier transform is established.  相似文献   

8.
We prove the differentiability of generalized Fourier transforms associated with a self-adjoint and strictly elliptic perturbation A of the Laplacian with variable coefficients in an exterior domain, using results on the spectral differentiability of the resolvent of A. Moreover we show that differentiable functions with bounded support and vanishing near the origin are mapped by the generalized Fourier transform into polynomially weighted L 2-spaces. As an application of the generalized Fourier transform and exploiting the previous results, we deal with equations of Kirchhoff type. We will not only show the global (in t) existence and uniqueness of solutions for a class of small data, but also an assertion on its time asymptotic behavior. In addition, we obtain amplified results for Schr?dinger operators . Received March 1999  相似文献   

9.
The unified transform method introduced by Fokas can be used to analyze initial‐boundary value problems for integrable evolution equations. The method involves several steps, including the definition of spectral functions via nonlinear Fourier transforms and the formulation of a Riemann‐Hilbert problem. We provide a rigorous implementation of these steps in the case of the mKdV equation in the quarter plane under limited regularity and decay assumptions. We give detailed estimates for the relevant nonlinear Fourier transforms. Using the theory of L2‐RH problems, we consider the construction of quarter plane solutions which are C1 in time and C3 in space.  相似文献   

10.
Weighted norm inequalities for the Fourier transform have a long history, motivated in part by generalizations of the differentiation formula for Fourier transforms and by applications such as Kolmogorov′s prediction theory and restriction theory. Typically, a norm inequality, || ||Lqμ ≤ ||f||Lpv, is established on a subspace of L1 contained in the weighted space Lpv, where is the Fourier transform and μ and v are weights. The problem of defining the extension of on all of Lpv is posed and solved here for several important cases. This definition sometimes results in the ordinary distributional Fourier transform, but there are other situations which are also analyzed. Precise necessary conditions and closure theorems are inextricably related to this program, and these results are established.  相似文献   

11.
We obtain a class of subsets of R2d such that the support of the short time Fourier transform (STFT) of a signal fL2(Rd) with respect to a window gL2(Rd) cannot belong to this class unless f or g is identically zero. Moreover we prove that the L2-norm of the STFT is essentially concentrated in the complement of such a set. A generalization to other Hilbert spaces of functions or distributions is also provided. To this aim we obtain some results on compactness of localization operators acting on weighted modulation Hilbert spaces.  相似文献   

12.
A sufficient condition of Pólya type is given for a function to be the Fourier transform of a unimodal distribution. This condition is used to show that exp(− ¦ x ¦3 − 3 ¦ x ¦) is the Fourier transform of a unimodal distribution.  相似文献   

13.
We address the problem of optimal reconstruction of the values of a linear operator on ℝ d or ℤ d from approximate values of other operators. Each operator acts as the multiplication of the Fourier transform by a certain function. As an application, we present explicit expressions for optimal methods of reconstructing the solution of the heat equation (for continuous and difference models) at a given instant of time from inaccurate measurements of this solution at other time instants.  相似文献   

14.
A fast transform for spherical harmonics   总被引:2,自引:0,他引:2  
Spherical harmonics arise on the sphere S2 in the same way that the (Fourier) exponential functions {eik}k arise on the circle. Spherical harmonic series have many of the same wonderful properties as Fourier series, but have lacked one important thing: a numerically stable fast transform analogous to the Fast Fourier Transform (FFT). Without a fast transform, evaluating (or expanding in) spherical harmonic series on the computer is slow—for large computations probibitively slow. This paper provides a fast transform.For a grid ofO(N2) points on the sphere, a direct calculation has computational complexityO(N4), but a simple separation of variables and FFT reduce it toO(N3) time. Here we present algorithms with timesO(N5/2 log N) andO(N2(log N)2). The problem quickly reduces to the fast application of matrices of associated Legendre functions of certain orders. The essential insight is that although these matrices are dense and oscillatory, locally they can be represented efficiently in trigonometric series.  相似文献   

15.
In this paper we present algorithms to calculate the fast Fourier synthesis and its adjoint on the rotation group SO(3) for arbitrary sampling sets. They are based on the fast Fourier transform for nonequispaced nodes on the three-dimensional torus. Our algorithms evaluate the SO(3) Fourier synthesis and its adjoint, respectively, of B-bandlimited functions at M arbitrary input nodes in O(M+B4)\mathcal O(M+B^4) or even O(M + B3 log2 B)\mathcal O(M + B^3 \log^2 B) flops instead of O(MB3)\mathcal O(MB^3). Numerical results will be presented establishing the algorithm’s numerical stability and time requirements.  相似文献   

16.
We give a criterion for the denseness of the Fourier transform in the space L 2 associated with spectral representation of positive-definite Toeplitz kernels.  相似文献   

17.
《偏微分方程通讯》2013,38(11-12):2267-2303
We prove a weighted L estimate for the solution to the linear wave equation with a smooth positive time independent potential. The proof is based on application of generalized Fourier transform for the perturbed Laplace operator and a finite dependence domain argument. We apply this estimate to prove the existence of global small data solution to supercritical semilinear wave equations with potential.  相似文献   

18.
Error estimates for scattered data interpolation by “shifts” of a conditionally positive definite function (CPD) for target functions in its native space, which is its associated reproducing kernel Hilbert space (RKHS), have been known for a long time. Regardless of the underlying manifold, for example ℝn or S n, these error estimates are determined by the rate of decay of the Fourier transform (or Fourier series) of the CPD. This paper deals with the restriction of radial basis functions (RBFs), which are radial CPD functions on ℝn+1, to the unit sphere S n. In the paper, we first strengthen a result derived by two of us concerning an explicit representation of the Fourier–Legendre coefficients of the restriction in terms of the Fourier transform of the RBF. In addition, for RBFs that are related to completely monotonic functions, we derive a new integral representation for these coefficients in terms of the measure generating the completely monotonic function. These representations are then utilized to show that if an RBF has a native space equivalent to a Sobolev space H s(ℝn+1), then the restriction to S n has a native space equivalent to H s−1/2(S n). In addition, they are used to recover the asymptotic behavior of such coefficients for a wide variety of RBFs. Some of these were known earlier. Joseph D. Ward: Francis J. Narcowich: Research supported by grant DMS-0204449 from the National Science Foundation.  相似文献   

19.
We characterize the radial functionsVin nfor which the a priori inequality u1/22+k uV1/22C V−1/2(Δ+k2) u2holds with constant independent ofk. The condition is forVto have the X-rays transform everywhere bounded. We apply these estimates to the well posedness of evolution Schrödinger equations with time dependent drift terms and to the restriction of the Fourier transform to Euclidean spheres.  相似文献   

20.
We prove estimates of the L2 norms on relatively dense subsets of the real line of functions with Fourier transforms supported on lacunary sets of intervals. This result is a real line analogue of Zygmund's theorem on lacunary trigonometric series. The results also hold in higher dimensions.  相似文献   

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